The Birman-Schwinger principle was established by Birman [1]} and Schwinger [2]} for Schrödinger operators \(\Delta +V\)  on \(\mathbb {R}^n\) , \(n\ge 3\) . It is the main impetus for using Cwikel estimates to establish the Cwikel-Lieb-Rozenblum inequality (see [3]}, [4]}). In our setting it relates the counting function
\(N(H_{V};\lambda )\)  to the eigenvalues of the Birman-Schwinger operators,
\(K_V(\lambda )=-(H-\lambda )^{-\frac{1}{2}}V(H-\lambda )^{-\frac{1}{2}},\qquad \lambda <0.\) 
